Preservation and reflection in specification

We extend the traditional notion of specification based on theories and interpretations between theories to model situations, typical of open, reactive systems, in which properties exhibited locally by an object no longer hold when that object is interconnected as a component of a larger system. The proposed notion of specification is based on the observation, due to Winskel, that while some assertions are preserved across morphisms of labelled transition systems, other are reflected. The distinction between these two classes of assertions leads us to the definition of two categories of specifications, one that supports horizontal structuring and another that supports vertical structuring, for which compositionality is proved.